INFORMATION ON THE IMAGE IN PSL(2,Z) OF THE GROUP OF PURE MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE Index in PSL(2,Z): 144 Minimal number of generators: 25 Number of equivalence classes of cusps: 20 Genus: 3 REPRESENTATIVES OF THE CUSP EQUIVALENCE CLASSSES -1/1 -4/9 -1/3 -2/7 0/1 1/4 1/3 3/7 1/2 2/3 1/1 11/9 3/2 2/1 7/3 8/3 3/1 4/1 6/1 1/0 CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE PSEUDOIMAGES -1/1 0/1 -1/2 0/1 1/2 -4/9 0/1 1/2 1/1 -3/7 1/1 -5/12 1/2 1/1 -2/5 0/1 1/1 -1/3 1/1 -2/7 0/1 1/1 1/0 -3/11 1/1 -1/4 1/1 1/0 -2/9 -2/1 1/0 -1/5 -1/1 0/1 0/1 1/0 1/5 -1/1 1/4 0/1 2/7 0/1 1/2 1/3 1/1 4/11 1/1 2/1 1/0 3/8 2/1 1/0 2/5 2/1 1/0 3/7 1/0 4/9 -4/1 1/0 1/2 -1/1 1/0 2/3 -1/1 0/1 1/0 3/4 -1/1 1/0 7/9 -1/1 4/5 0/1 1/0 1/1 -1/1 6/5 -2/3 -1/2 11/9 -1/2 5/4 -1/2 0/1 4/3 -1/2 0/1 3/2 0/1 8/5 0/1 1/0 13/8 0/1 1/0 5/3 -1/1 2/1 -1/1 0/1 7/3 -1/1 12/5 -1/1 -2/3 5/2 -1/1 -1/2 13/5 -1/1 21/8 -2/3 -1/2 29/11 -1/2 8/3 -1/2 0/1 11/4 0/1 3/1 -1/1 4/1 -1/1 -1/2 0/1 5/1 -1/1 6/1 -1/2 0/1 1/0 -1/2 0/1 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,-2,-3) (-1/1,1/0) -> (-1/1,-1/2) Parabolic Matrix(35,16,94,43) (-1/2,-4/9) -> (4/11,3/8) Hyperbolic Matrix(37,16,104,45) (-4/9,-3/7) -> (1/3,4/11) Hyperbolic Matrix(105,44,136,57) (-3/7,-5/12) -> (3/4,7/9) Hyperbolic Matrix(169,70,70,29) (-5/12,-2/5) -> (12/5,5/2) Hyperbolic Matrix(31,12,18,7) (-2/5,-1/3) -> (5/3,2/1) Hyperbolic Matrix(27,8,-98,-29) (-1/3,-2/7) -> (-2/7,-3/11) Parabolic Matrix(107,28,42,11) (-3/11,-1/4) -> (5/2,13/5) Hyperbolic Matrix(25,6,54,13) (-1/4,-2/9) -> (4/9,1/2) Hyperbolic Matrix(75,16,14,3) (-2/9,-1/5) -> (5/1,6/1) Hyperbolic Matrix(1,0,10,1) (-1/5,0/1) -> (0/1,1/5) Parabolic Matrix(67,-14,24,-5) (1/5,1/4) -> (11/4,3/1) Hyperbolic Matrix(109,-30,40,-11) (1/4,2/7) -> (8/3,11/4) Hyperbolic Matrix(61,-18,78,-23) (2/7,1/3) -> (7/9,4/5) Hyperbolic Matrix(129,-50,80,-31) (3/8,2/5) -> (8/5,13/8) Hyperbolic Matrix(43,-18,98,-41) (2/5,3/7) -> (3/7,4/9) Parabolic Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(217,-262,82,-99) (6/5,11/9) -> (29/11,8/3) Hyperbolic Matrix(305,-376,116,-143) (11/9,5/4) -> (21/8,29/11) Hyperbolic Matrix(27,-34,4,-5) (5/4,4/3) -> (6/1,1/0) Hyperbolic Matrix(25,-36,16,-23) (4/3,3/2) -> (3/2,8/5) Parabolic Matrix(167,-274,64,-105) (13/8,5/3) -> (13/5,21/8) Hyperbolic Matrix(43,-98,18,-41) (2/1,7/3) -> (7/3,12/5) Parabolic Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS UNDER THE VIRTUAL ENDOMORPHISM Matrix(1,2,-2,-3) -> Matrix(1,0,4,1) Matrix(35,16,94,43) -> Matrix(3,-2,2,-1) Matrix(37,16,104,45) -> Matrix(3,-2,2,-1) Matrix(105,44,136,57) -> Matrix(1,0,-2,1) Matrix(169,70,70,29) -> Matrix(3,-2,-4,3) Matrix(31,12,18,7) -> Matrix(1,0,-2,1) Matrix(27,8,-98,-29) -> Matrix(1,0,0,1) Matrix(107,28,42,11) -> Matrix(1,0,-2,1) Matrix(25,6,54,13) -> Matrix(1,-2,0,1) Matrix(75,16,14,3) -> Matrix(1,2,-2,-3) Matrix(1,0,10,1) -> Matrix(1,0,0,1) Matrix(67,-14,24,-5) -> Matrix(1,0,0,1) Matrix(109,-30,40,-11) -> Matrix(1,0,-4,1) Matrix(61,-18,78,-23) -> Matrix(1,0,-2,1) Matrix(129,-50,80,-31) -> Matrix(1,-2,0,1) Matrix(43,-18,98,-41) -> Matrix(1,-6,0,1) Matrix(13,-8,18,-11) -> Matrix(1,0,0,1) Matrix(11,-10,10,-9) -> Matrix(1,2,-2,-3) Matrix(217,-262,82,-99) -> Matrix(3,2,-8,-5) Matrix(305,-376,116,-143) -> Matrix(5,2,-8,-3) Matrix(27,-34,4,-5) -> Matrix(1,0,0,1) Matrix(25,-36,16,-23) -> Matrix(1,0,2,1) Matrix(167,-274,64,-105) -> Matrix(1,2,-2,-3) Matrix(43,-98,18,-41) -> Matrix(1,2,-2,-3) Matrix(9,-32,2,-7) -> Matrix(1,0,0,1) INFORMATION ON THE IMAGE OF THIS GROUP UNDER THE VIRTUAL ENDOMORPHISM Index in PSL(2,Z): 6 Minimal number of generators: 2 Number of equivalence classes of cusps: 3 Genus: 0 Degree of H/liftables -> H/(image of liftables): 5 Degree of the the map X: 5 Degree of the the map Y: 24 Permutation triple for Y: ((1,6,18,7,2)(3,11,21,8,4)(10,17,14,24,23)(12,16,15,19,13); (1,4,13,14,5)(3,9,19,18,10)(6,16,22,21,17)(7,20,23,12,8); (1,2,8,22,16,23,24,13,9,3)(4,12)(5,14,21,11,10,20,7,19,15,6)(17,18)) ----------------------------------------------------------------------- Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift elements of DeckMod(f) via pi_1: 0 DeckMod(f) is trivial. Elements among 0, lambda1, lambda2 and lambda1+lambda2 which lift modular group liftables via pi_1: 0 The subgroup of modular group liftables which arise from translations is trivial. ----------------------------------------------------------------------- The image of the modular group liftables in PSL(2,Z) equals the image of the pure modular group liftables. ----------------------------------------------------------------------- INFORMATION ON THE IMAGE IN PGL(2,Z) OF THE GROUP OF EXTENDED MODULAR GROUP LIFTABLES UNDER ITS PULLBACK ACTION ON THE UPPER HALF-PLANE CUSPS AT THE FUNDAMENTAL DOMAIN AND THEIR IMAGES UNDER THE PULLBACK MAP CUSP IMAGE c d -1/1 0/1 2 1 0/1 (0/1,1/0) 0 10 1/4 0/1 2 2 2/7 (0/1,1/2) 0 10 1/3 1/1 1 5 3/8 (2/1,1/0) 0 10 2/5 (2/1,1/0) 0 10 3/7 1/0 3 1 1/2 (-1/1,1/0) 0 10 2/3 0 2 3/4 (-1/1,1/0) 0 10 4/5 (0/1,1/0) 0 10 1/1 -1/1 1 5 6/5 (-2/3,-1/2) 0 10 11/9 -1/2 2 1 5/4 (-1/2,0/1) 0 10 4/3 (-1/2,0/1) 0 10 3/2 0/1 1 2 2/1 (-1/1,0/1) 0 10 7/3 -1/1 1 1 5/2 (-1/1,-1/2) 0 10 8/3 (-1/2,0/1) 0 10 3/1 -1/1 1 5 4/1 0 2 5/1 -1/1 1 5 1/0 (-1/2,0/1) 0 10 GENERATING SET ASSOCIATED TO THE FUNDAMENTAL DOMAIN GENERATOR EDGE PAIRING TYPE Matrix(1,2,0,-1) (-1/1,1/0) -> (-1/1,1/0) Reflection Matrix(-1,0,2,1) (-1/1,0/1) -> (-1/1,0/1) Reflection Matrix(1,0,8,-1) (0/1,1/4) -> (0/1,1/4) Reflection Matrix(15,-4,56,-15) (1/4,2/7) -> (1/4,2/7) Reflection Matrix(45,-14,16,-5) (2/7,1/3) -> (8/3,3/1) Glide Reflection Matrix(43,-16,8,-3) (1/3,3/8) -> (5/1,1/0) Glide Reflection Matrix(57,-22,44,-17) (3/8,2/5) -> (5/4,4/3) Glide Reflection Matrix(29,-12,70,-29) (2/5,3/7) -> (2/5,3/7) Reflection Matrix(13,-6,28,-13) (3/7,1/2) -> (3/7,1/2) Reflection Matrix(13,-8,18,-11) (1/2,2/3) -> (2/3,3/4) Parabolic Matrix(57,-44,22,-17) (3/4,4/5) -> (5/2,8/3) Glide Reflection Matrix(11,-10,10,-9) (4/5,1/1) -> (1/1,6/5) Parabolic Matrix(109,-132,90,-109) (6/5,11/9) -> (6/5,11/9) Reflection Matrix(89,-110,72,-89) (11/9,5/4) -> (11/9,5/4) Reflection Matrix(17,-24,12,-17) (4/3,3/2) -> (4/3,3/2) Reflection Matrix(7,-12,4,-7) (3/2,2/1) -> (3/2,2/1) Reflection Matrix(13,-28,6,-13) (2/1,7/3) -> (2/1,7/3) Reflection Matrix(29,-70,12,-29) (7/3,5/2) -> (7/3,5/2) Reflection Matrix(9,-32,2,-7) (3/1,4/1) -> (4/1,5/1) Parabolic IMAGES OF THE GENERATORS MAP ON REFLECTION AXES OR UNDER THE VIRTUAL ENDOMORPHISM FIXED POINT OF IMAGE Matrix(1,2,0,-1) -> Matrix(-1,0,4,1) (-1/1,1/0) -> (-1/2,0/1) Matrix(-1,0,2,1) -> Matrix(1,0,0,-1) (-1/1,0/1) -> (0/1,1/0) Matrix(1,0,8,-1) -> Matrix(1,0,0,-1) (0/1,1/4) -> (0/1,1/0) Matrix(15,-4,56,-15) -> Matrix(1,0,4,-1) (1/4,2/7) -> (0/1,1/2) Matrix(45,-14,16,-5) -> Matrix(1,0,0,-1) *** -> (0/1,1/0) Matrix(43,-16,8,-3) -> Matrix(1,-2,-2,3) Matrix(57,-22,44,-17) -> Matrix(1,-2,-2,3) Matrix(29,-12,70,-29) -> Matrix(-1,4,0,1) (2/5,3/7) -> (2/1,1/0) Matrix(13,-6,28,-13) -> Matrix(1,2,0,-1) (3/7,1/2) -> (-1/1,1/0) Matrix(13,-8,18,-11) -> Matrix(1,0,0,1) Matrix(57,-44,22,-17) -> Matrix(-1,0,2,1) *** -> (-1/1,0/1) Matrix(11,-10,10,-9) -> Matrix(1,2,-2,-3) -1/1 Matrix(109,-132,90,-109) -> Matrix(7,4,-12,-7) (6/5,11/9) -> (-2/3,-1/2) Matrix(89,-110,72,-89) -> Matrix(-1,0,4,1) (11/9,5/4) -> (-1/2,0/1) Matrix(17,-24,12,-17) -> Matrix(-1,0,4,1) (4/3,3/2) -> (-1/2,0/1) Matrix(7,-12,4,-7) -> Matrix(-1,0,2,1) (3/2,2/1) -> (-1/1,0/1) Matrix(13,-28,6,-13) -> Matrix(-1,0,2,1) (2/1,7/3) -> (-1/1,0/1) Matrix(29,-70,12,-29) -> Matrix(3,2,-4,-3) (7/3,5/2) -> (-1/1,-1/2) Matrix(9,-32,2,-7) -> Matrix(1,0,0,1) ----------------------------------------------------------------------- The pullback map was not drawn because it is too complicated.